07 - balance principles. me338 - syllabus balance principles balance principles. cauchy s postulate. cauchy s lemma -t

Transcription

1 me338 - syllabus holzapfel nonlinear solid mechanics [2000], chapter 4, pages cauchy s postulate cauchy s lemma cauchy s postulate stress vector t to a plane with normal n at position x only depends on plane s normal n Px cauchy s lemma -t -n newton s third law actio = reactio n t 3 4

2 cauchy s theorem cauchy X 3, x 3 cauchy s postulate t 2 t t 1 X 2, x 2 stress vector t to a plane with normal n at position x only depends on plane s normal n cauchy s lemma X 1, x 1 t 3 cauchy s theorem newton s third law actio = reactio cauchy s theorem existence of second order tensor field σ is inde- pendent of n, such that t is a linear function of n existence of second order tensor field σ is inde- pendent of n, such that t is a linear function of n 5 6 illustration of stress components normal and tangential stress x 3 e 3 stress vector t n n t interpretation of 3x3 components normal stress t t e 2 x 2 tangential stress x 1 e 1 7 8

3 stress tensors cauchy / true stress relates spatial force to spatial area first piola kirchhoff / nominal stress relates spatial force to material area second piola kirchhoff stress relates material force to material area stress tensors cauchy / true stress relates spatial force to spatial area first piola kirchhoff / nominal stress relates spatial force to material area second piola kirchhoff stress relates material force to material area 9 10 stress tensors transport mechanisms covariant / strains first piola kirchhoff pull back push forward gustav robert kirchhoff [ ] second piola kirchhoff cauchy contravariant / stresses pull back push forward 11 12

4 of mass, momentum, angular momentum and energy, supplemented with an entropy inequality constitute the set of conservation laws. the law of conservation of mass/matter states that the mass of a closed system of substances will remain constant, regardless of the processes acting inside the system. the principle of conservation of momentum states that the total momentum of a closed system of objects is constant. of mass, linear momentum, angular momentum and energy apply to all material bodies. each one gives rise to a field equation, holing on the configurations of a body in a sufficiently smooth motion and a jump condition on surfaces of discontinuity. like position, time and body, the concepts of mass, force, heating and internal energy which enter into the formulation of the balance equations are regarded as having primitive status in continuum mechanics. chadwick continuum mechanics [1976] potato potato [1] isolate subset from [1] isolate subset from [2] characterize influence of remaining body through phenomenological quantities - contact fluxes, &

5 potato potato [1] isolate subset [2] characterize influence of remaining body through phenomenological quantities - contact fluxes, & [3] from define basic physical quantities - mass, linear and angular momentum, energy generic balance equation [1] isolate subset from [2] characterize influence of remaining body through phenomenological quantities - contact fluxes, & [3] define basic physical quantities - mass, linear and angular momentum, energy [4] postulate balance of these quantities here: closed systems general format balance quantity flux source production unlike open systems closed systems have a constant mass examples of open systems: rocket propulsion and biological growth (me337)

6 transformation of volume elements - determinant of mass is always constant and positive changes in volume - determinant of deformation tensor changes in volume and density are related through and mass is constant global material density is constant local

7 balance of (linear) momentum density no mass flux no mass source no mass production continuity equation linear momentum density momentum flux - stress momentum source - force no momentum production equilibrium equation compare balance of (internal) energy internal energy density heat flux heat source no heat production mass point energy equation internal mechanical power external thermal power